非自治Kadomtsev-Petviashvili方程的自相似变换和二维怪波构造

张解放; 金美贞; 胡文成

doi:10.7498/aps.69.20200981

摘要

首先给出非自治Kadomtsev-Petviashvili方程转换为Kadomtsev- Petviashvili方程的一个自相似变换, 然后基于Kadomtsev-Petviashvili方程的Lump解构造了非自治Kadomtsev-Petviashvili方程的有理函数表示的二维单、双、三怪波解, 最后通过合适选取变参数, 用图示说明了它们的演化特征, 并利用快速傅里叶变换算法数值模拟测试了二维单怪波的动力学稳定性. 本文方法对寻找(2 + 1)维非线性波动模型的怪波激发提供了启迪.

关键词:

Abstract

Rogue wave is a kind of natural phenomenon that is fascinating, rare, and extreme. It has become a frontier of academic research. The rogue wave is considered as a spatiotemporal local rational function solution of nonlinear wave model. There are still very few (2 + 1)-dimensional nonlinear wave models which have rogue wave solutions, in comparison with soliton and Lump waves that are found in almost all (2 + 1)-dimensional nonlinear wave models and can be solved by different methods, such as inverse scattering method, Hirota bilinear method, Darboux transform method, Riemann-Hilbert method, and homoclinic test method. The structure and evolution characteristics of the obtained (2 + 1)-dimensional rogue waves are quite different from the prototypes of the (1 + 1)-dimensional nonlinear Schrödinger equation. Therefore, it is of great value to study two-dimensional rogue waves.In this paper, the non-autonomous Kadomtsev-Petviashvili equation is first converted into the Kadomtsev-Petviashvili equation with the aid of a similar transformation, then two-dimensional rogue wave solutions represented by the rational functions of the non-autonomous Kadomtsev-Petviashvili equation are constructed based on the Lump solution of the first kind of Kadomtsev-Petviashvili equation, and their evolutionary characteristics are illustrated by images through appropriately selecting the variable parameters and the dynamic stability of two-dimensional single rogue waves is numerically simulated by the fast Fourier transform algorithm. The obtained two-dimensional rogue waves, which are localized in both space and time, can be viewed as a two-dimensional analogue to the Peregrine soliton and thus are a natural candidate for describing the rogue wave phenomena. The method presented here provides enlightenment for searching for rogue wave excitation of (2 + 1)-dimensional nonlinear wave models.We show that two-dimensional rogue waves are localized in both space and time which arise from the zero background and then disappear into the zero background again. These rogue-wave solutions to the non-autonomous Kadomtsev-Petviashvili equation generalize the rogue waves of the nonlinear Schrödinger equation into two spatial dimensions, and they could play a role in physically understanding the rogue water waves in the ocean.

Keywords:

作者及机构信息

1.
浙江传媒学院智能媒体技术研究院, 杭州　310018

2.
浙江传媒学院, 浙江省影视媒体技术研究重点实验室, 杭州　310018

3.
浙江传媒学院网络数据中心, 杭州　310018

4.
中原工学院理学院, 郑州　450007

通信作者: 张解放, Zhangjief@cuz.edu.cn

基金项目: 国家自然科学基金(批准号: 61877053)资助的课题

Authors and contacts

1.
Institute of Intelligent Media Technology, Communication University of Zhejiang, Hangzhou 310018, China

2.
Zhejiang Provincial Key Laboratory of Film and Television Media Technology, Communication University of Zhejiang, Hangzhou 310018, China

3.
Network Data Center, Communication University of Zhejiang, Hangzhou 310018, China

4.
College of Science, Zhongyuan University of Technology, Zhengzhou 450007, China

Corresponding author: Zhang Jie-Fang, Zhangjief@cuz.edu.cn

Funds: Project supported by the National Natural Science Foundation of China (Grant No. 61877053)

文章全文 : translate this paragraph

参考文献

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施引文献

图 1 由(12)式所确定的非自治KP方程的二维单怪波演化　(a) $t \!=\! - 6$; (b)$t \!=\! - 3$; (c) $t \!=\! 0$; (d) $t \!=\! 0.5$; (e) $t \!=\! 4$; (f) $t \!=\! 8$

Fig. 1. Evolution of two-dimensional single rogue wave propagation given in Eq. (12) for non-autonomous KP equation: (a)$t = - 6$; (b) $t = - 3$; (c) $t = 0$; (d) $t = 0.5$; (e) $t = 4$; (f) $t = 8$.

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图 2 由(13)式所确定的非自治KP方程的二维单怪波演化　(a)$t = - 6$; (b)$t = - 3$; (c)$t = 0$; (d)$t = 0.5$; (e)$t = 4$; (f)$t = 8$

Fig. 2. Evolution of two-dimensional single rogue wave propagation given in Eq. (13) for non-autonomous KP equation: (a)$t = - 6$; (b)$t = - 3$; (c)$t = 0$; (d)$t = 0.5$; (e)$t = 4$; (f) $t = 8$.

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图 3 由(14)式所确定的非自治KP方程的二维单怪波演化　(a)$t = - 6$; (b) $t = - 3$; (c)$t = 0$; (d) $t = 0.5$; (e)$t = 4$; (f)$t = 8$

Fig. 3. Evolution of two-dimensional single rogue wave propagation given in Eq. (14) for non-autonomous KP equation: (a)$t = - 6$; (b) $t = - 3$; (c)$t = 0$; (d)$t = 0.5$; (e)$t = 4$; (f)$t = 8$.

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图 4 由(11)式所确定的非自治KP方程的二维双怪波演化(选取$k = 1/2, l = 1/2, n = 0, m = 1, \lambda = \varepsilon = 1, \nu = 1, \chi = 0, {\upsilon _x} = 2, {\upsilon _y} = 1, a = b = 0$)　(a)$t = - 6$; (b)$t = - 3$; (c)$t = 0$; (d) $t = 0.5$; (e)$t = 4$; (f)$t = 8$

Fig. 4. Time evolution of two-dimensional double rogue waves propagation given in Eq. (11) for non-autonomous KP equation when$k = 1/2, l = 1/2, n = 0, m = 1, $ $\lambda = \varepsilon = 1, \nu = 1, \chi = 0, {\upsilon _x} = 2, {\upsilon _y} = 1, a = b = 0$: (a)$t = - 6$; (b)$t = - 3$; (c)$t = 0$; (d) $t = 0.5$; (e)$t = 4$; (f)$t = 8$.

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图 5 由(11)式所确定的非自治KP方程二维三怪波演化(选择$k = l = 1/2, n = 0, m = 1, $ $\lambda = \varepsilon = 1, \nu = 1, \chi = 0, {\upsilon _x} = 2, {\upsilon _y} = 2, a = 5000, b = 5000$)　(a) $t = - 6$; (b) $t = - 3$; (c) $t = 0$; (d) $t = 0.5$; (e) $t = 4$; (f) $t = 8$

Fig. 5. Time evolution of two-dimensional triple rogue waves propagation given in Eq. (11) for non-autonomous KP equation when $k = l = 1/2, n = 0, m = 1, \lambda = \varepsilon = 1, $ $\nu = 1, \chi = 0, {\upsilon _x} = 2, {\upsilon _y} = 1, a = 5000, b = 5000$: (a) $t = - 6$; (b) $t = - 3$; (c) $t = 0$; (d) $t = 0.5$; (e) $t = 4$; (f) $t = 8$.

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图 6 由(11)式所确定的二维双、三怪波(选取$k = 1/2, l = 1/2, n = 0, m = 1, $ $\lambda = 1, \varepsilon = 1, \nu = 1, \chi = 0, {\upsilon _x} = 2$, $\mu =0, \mu =0.8165, \mu =-0.8165$分别对应选取${\upsilon _y} = 0, {\upsilon _y} = - 2, {\upsilon _y} = 2$)

Fig. 6. Profiles of two-dimensional double and triple rogue waves given in Eq. (11) for non-autonomous KP equation when $k = 1/2, l = 1/2, n = 0, m = 1, \lambda = 1, \varepsilon = 1, \nu = 1, $ $\chi = 0, {\upsilon _x} = 2$,$\mu =0, \mu =0.8165, \mu =-0.8165$ correspond to ${\upsilon _y}{{ = 0}}, {\upsilon _y} = - 2, {\upsilon _y} = 2$, respectively.

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图 7 加了高斯白噪声扰动后由(15)式所确定的二维单怪波演化　(a) $t = - 5$; (b) $t = - {\rm{3}}$; (c) $t = - 1$; (d) $t = 0$; (e) $t = {2}.5$; (f) $t = 4$

Fig. 7. Evolution of two-dimensional single rogue wave determined by Eq. (15) after Gaussian white noise disturbance: (a) $t = - 5$; (b) $t = - {\rm{3}}$; (c) $t = - 1$; (d) $t = 0$; (e) $t = {2}.5$; (f) $t = 4$.

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图 8 加了高斯白噪声扰动后由(16)式所确定的二维单怪波演化　(a) $t = - 5$; (b) $t = - {\rm{3}}$, (c) $t = - 1$; (d) $t = 0$; (e) $t = {2}.5$; (f) $t = 4$

Fig. 8. Evolution of two-dimensional single rogue wave determined by Eq. (16) after Gaussian white noise disturbance: (a) $t = - 5$; (b) $t = - {\rm{3}}$; (c) $t = - 1$; (d) $t = 0$; (e) $t = {2}.5$; (f) $t = 4$.

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图 9 在时间区间[–5, 5] x-y平面上非自治KP方程的二维单怪波最大波动值和最小波动值的解析结果和数值计算模拟的对照图　(a)对应二维单怪波((15)式); (b)对应二维单怪波((16)式); (c)在(a)中加了高斯白噪声扰动; (d)在(b)中加高斯白噪声扰动

Fig. 9. Simulation diagram of the analytic and numerical results of the maximum and minimum fluctuations of two-dimensional single rogue waves for the non- autonomous KP equation in the x-y plane of the time interval [–5, 5]: (a) Corresponds to a two-dimensional single rogue wave (Eq. (15)); (b) Corresponds to a two- dimensional single rogue wave (Eq. (16)); (c) Gaussian white noise is added in panel (a); (d) Gaussian white noise is added in panel (b).

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map

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[12]	Ganshin A N, Efimov V B, Kolmakov G V, Mezhov Deglin P V, McClintock E 2008 Phys. Rev. Lett. 101 065303 Google Scholar
[13]	Yan Z Y 2010 Commun. Theor. Phys. 54 947 Google Scholar
[14]	Shats M, Punzmann H, Xia H 2010 Phys. Rev. Lett. 104 104503 Google Scholar
[15]	Xia H, Maimbourg T, Punzmann H, Shats M 2012 Phys. Rev. Lett. 109 114502 Google Scholar
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[17]	Chabchoub A, Hoffmann N P, Akhmediev N 2011 Phys. Rev. Lett. 106 204502 Google Scholar
[18]	Peregrine D H 1983 J. Aust. Math. Soc. Ser. B: Appl. Math. 25 16 Google Scholar
[19]	Akhmediev N, Ankiewicz A, Soto Crespo J M 2009 Phys. Rev. E 80 026601 Google Scholar
[20]	Kedziora D J, Ankiewicz A, Akhmediev N 2012 Phys. Rev. E 86 056602 Google Scholar
[21]	Ohta Y, Yang J 2012 Proc. R. Soc. A 468 1716 Google Scholar
[22]	Ankiewicz A, Soto Crespo J M, Akhmediev N 2010 Phys. Rev. E 81 046602 Google Scholar
[23]	Li L J, Wu Z W, Wang J H, He J S 2013 Annals of Physics 334 198 Google Scholar
[24]	Tao Y S, He J S 2012 Phys. Rev. E 85 026601 Google Scholar
[25]	Chen S 2013 Phys. Rev. E 88 023202 Google Scholar
[26]	Chan H N, Chow K W, Kedziora D J, Grimshaw R H J, Ding E 2014 Phys. Rev. E 89 032914 Google Scholar
[27]	Zhang Y S, Guo L J, He J S 2015 Lett. Math. Phys. 105 853 Google Scholar
[28]	Qiu D Q, He J, Zhang Y H, Porsezian K 2015 Proc. R. Soc. A 471 20150236 Google Scholar
[29]	He J S, Xu S W, Porsezian K 2012 J. Phs. Soc. Japan 81 124007 Google Scholar
[30]	Xu S W, He J S, Cheng Y, Porseizan K 2015 Math. Meth. Appli. Sci. 38 1106 Google Scholar
[31]	Chen S, Song L Y 2014 Phys. Lett. A 378 1228 Google Scholar
[32]	He J S, Wang L, Li L, Porsezian K, Erdélyi R 2014 Phys. Rev. E 89 062917 Google Scholar
[33]	Zha Q 2013 Phys. Scr. 87 065401 Google Scholar
[34]	Chen S, Soto Crespo J M, Baronio F, Grelu Ph, Mihalache D 2016 Opt. Express 24 15251 Google Scholar
[35]	Wang L H, Porsezian K, He J S 2013 Phys. Rev. E 87 053202 Google Scholar
[36]	Chen S, Mihalache D 2015 J. Phys. A: Math. Theor. 48 215202 Google Scholar
[37]	Baronio F, Conforti M, Degasperis A, Lombardo S, Onorato M, Wabnitz S 2014 Phys. Rev. Lett. 113 034101 Google Scholar
[38]	He J S, Zhang H R, Wang L H, Porsezian K, Fokas A S 2013 Phys. Rev. E 87 052914 Google Scholar
[39]	Wang L H, He J S, Xu H, Wang J, Porsezian K 2017 Phys. Rev. E 95 042217 Google Scholar
[40]	Ohta Y, Yang J 2012 Phys. Rev. E 86 036604 Google Scholar
[41]	Ohta Y, Yang J 2013 J. Phys. A: Math. Theor. 46 105202 Google Scholar
[42]	Rao J G, Porsezian K, He J S 2017 Chaos 27 083115 Google Scholar
[43]	Guo L J, He J S, Wang L H, Cheng Y, Frantzeskakis D J, Kevrekidis P G 2020 Phys. Rev. Res. 2 033376 Google Scholar
[44]	Wen L L, Zhang H Q 2016 Nonlinear Dyn. 86 877 Google Scholar
[45]	Qiu D Q, Zhang Y S, He J S 2016 Commun. Nonlinear Sci. Numer. Simulat. 30 307 Google Scholar
[46]	Jia R R, Guo R 2019 Appl. Math. Lett. 93 117 Google Scholar
[47]	Kadomtsev B B, Petviashvili V I 1970 Sov. Phys. Dokl. 15 539
[48]	Ablowitz M J, Segur H 1979 J. Fluid Mech. 92 691 Google Scholar
[49]	Pelinovsky D E, Stepanyants Y A, Kivshar Y A 1995 Phys. Rev. E 51 5016 Google Scholar
[50]	Manakov S V, Zakharov V E, Bordag L A, Matveev V B 1977 Phys. Lett. A 63 205 Google Scholar
[51]	Krichever I 1978 Funct. Anal. and Appl. 12 59
[52]	Satsuma J, Ablowitz M J 1979 J. Math. Phys. 20 1496 Google Scholar
[53]	Pelinovsky D E, Stepanyants Y A 1993 JETP Lett. 57 24
[54]	Pelinovsky D E 1994 J. Math. Phys. 35 5820 Google Scholar
[55]	Ablowitz M J, Villarroel J 1997 Phys. Rev. Lett. 78 570 Google Scholar
[56]	Villarroel J, Ablowitz M J 1999 Comm. Math. Phys. 207 1 Google Scholar
[57]	Biondini G, Kodama Y 2003 J. Phys. A: Math. Gen. 36 10519 Google Scholar
[58]	Kodama Y 2004 J. Phys. A: Math. Gen. 37 11169 Google Scholar
[59]	Biondini G 2007 Phys. Rev. Lett. 99 064103 Google Scholar
[60]	Ma W X 2015 Phys. Lett. A 379 1975 Google Scholar
[61]	Singh N, Stepanyants Y 2016 Wave Motion 64 92 Google Scholar
[62]	Hu W C, Huang W H, L u, Z M, Stepanyants Y 2018 Wave Motion 77 243 Google Scholar
[63]	Wen X Y, Yan Z Y 2017 Commun. Nonlinear Sci. Numer. Simulat. 43 311 Google Scholar
[64]	Yang J Y, Ma W X 2017 Nonlinear Dyn. 89 1539 Google Scholar
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非自治Kadomtsev-Petviashvili方程的自相似变换和二维怪波构造